Neighbour list object:
Number of regions: 543
Number of nonzero links: 3120
Percentage nonzero weights: 1.058169
Average number of links: 5.745856
Link number distribution:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 9 48 89 137 97 70 43 19 17 4 5 2 1 1
1 least connected region:
69 with 1 link
1 most connected region:
387 with 15 links
Unfortunately, we are yet done with creating the links between neighborhoods. What we receive is, in principle, a huge matrix with connected observations.
That’s nothing we could plug into a statistical model, such as a regression or the like (see next session).
Normalization
Normalization is the process of creating actual spatial weights. There is a huge dispute on how to do it (Neumayer & Plümper, 20161). But nobody questions whether it should be done in the first place since, among others, it restricts the parameter space of the weights.
One of the disputed but at the same time standard procedures is row-normalization. It divides all individual weights (=connections between spatial units) \(w_{ij}\) by the row-wise sum of of all other weights:
Characteristics of weights list object:
Neighbour list object:
Number of regions: 543
Number of nonzero links: 3120
Percentage nonzero weights: 1.058169
Average number of links: 5.745856
Link number distribution:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 9 48 89 137 97 70 43 19 17 4 5 2 1 1
1 least connected region:
69 with 1 link
1 most connected region:
387 with 15 links
Weights style: W
Weights constants summary:
n nn S0 S1 S2
W 543 294849 543 201.1676 2261.458
Most and foremost, Moran’s I makes use of the previously created weights between all spatial units pairs \(w_{ij}\). It weights deviations from an overall mean value of connected pairs according to the strength of the modeled spatial relations. Moran’s I can be interpreted as some sort of a correlation coefficient (-1 = perfect negative spatial autocorrelation; +1 = perfect positive spatial autocorrelation).
Moran I test under randomisation
data: election_results$immigrant_share
weights: queens_W
Moran I statistic standard deviate = 20.897, p-value < 2.2e-16
alternative hypothesis: greater
sample estimates:
Moran I statistic Expectation Variance
0.5398961097 -0.0018450185 0.0006720411
Test of spatial autocorrelation: Geary’s C
Moran’s I is a global statistic for spatial autocorrelation. It can produce issues when there are only local clusters of spatial interdependence in the data. An alternative is the use of Geary's C:
As you can see, in the numerator, the average value \(\bar{x}\) is not as prominent as in Moran’s I. Geary’s C only produces values between 0 and 2 (value near 0 = positive spatial autocorrelation; 1 = no spatial autocorrelation; values near 2 = negative spatial autocorrelation).
Geary C test under randomisation
data: election_results$immigrant_share
weights: queens_W
Geary C statistic standard deviate = 16.951, p-value < 2.2e-16
alternative hypothesis: Expectation greater than statistic
sample estimates:
Geary C statistic Expectation Variance
0.4649079513 1.0000000000 0.0009965167
Modern inferface to neighbors: sfdep package
The sfdep package provides a more tidyverse-compliant syntax to spatial weights. See:
Moran I test under randomisation
data: x
weights: listw
Moran I statistic standard deviate = 20.897, p-value < 2.2e-16
alternative hypothesis: greater
sample estimates:
Moran I statistic Expectation Variance
0.5398961097 -0.0018450185 0.0006720411
Geary C test under randomisation
data: x
weights: listw
Geary C statistic standard deviate = 16.951, p-value < 2.2e-16
alternative hypothesis: Expectation greater than statistic
sample estimates:
Geary C statistic Expectation Variance
0.4649079513 1.0000000000 0.0009965167
Measures of local spatial autocorrelation: LISA clusters
The reason why we show you the sfdep package is that it provides nice functions to calculate local measures of spatial autocorrelation. One popular choice are the estimation of Local Indicators of Spatial Autocorrelation (i.e., LISA clusters). In the most straightforward way they can be interpreted as case-specific indicators of spatial autocorrelation: